Geometric Reductions, Dynamics and Controls for Hamiltonian System with Symmetry

The paper states the Hamilton–Jacobi theorem for an RCH system on a cotangent bundle (Theorem 5.1) but defers its proof to prior work; its argument here is thus incomplete, though the statement matches standard identities based on Lemma 2.3 and the closed-loop vector field decomposition X̃ = X_H + vlift(F) + vlift(u) (eq. (3.1)) that ensures Tπ_Q ∘ X̃ = Tπ_Q ∘ X_H . The candidate solution correctly supplies the missing proof sketch: it uses γ*ω = −dγ, the verticality of X̃ − X_H, and symplectic covariance Tε ∘ X_{H∘ε} = X_H ∘ ε to derive (i) Tγ ∘ X̃^γ = X_H ∘ γ (and the classical equivalence to d(H∘γ)=0) and (ii) the Type II equivalence Tγ ∘ X̃^ε = X_H ∘ ε ⇔ Tε ∘ X_{H∘ε} = Tλ ∘ X̃ ∘ ε, aligning precisely with Theorem 5.1(ii) in the paper .